Problem: Simplify the following expression and state the condition under which the simplification is valid. $a = \dfrac{-6n^2 - 12n + 144}{-7n^2 + 84n - 224}$
First factor out the greatest common factors in the numerator and in the denominator. $ a = \dfrac {-6(n^2 + 2n - 24)} {-7(n^2 - 12n + 32)} $ $ a = \dfrac{6}{7} \cdot \dfrac{n^2 + 2n - 24}{n^2 - 12n + 32} $ Next factor the numerator and denominator. $ a = \dfrac{6}{7} \cdot \dfrac{(n - 4)(n + 6)}{(n - 4)(n - 8)}$ Assuming $n \neq 4$ , we can cancel the $n - 4$ $ a = \dfrac{6}{7} \cdot \dfrac{n + 6}{n - 8}$ Therefore: $ a = \dfrac{ 6(n + 6)}{ 7(n - 8)}$, $n \neq 4$